[Paper Review] New Sign Uncertainty Principles

Felipe Gonçalves, Diogo Oliveira e Silva+1 more|arXiv (Cornell University)|Mar 24, 2020

Mathematical functions and polynomials32 references4 citations

TL;DR

This paper introduces a generalized operator sign uncertainty principle that extends prior work on sign uncertainty in Fourier analysis, establishing quantitative lower bounds on the measure of negative regions for function-transform pairs across diverse operators. The key contribution is a universal inequality linking $L^p$ and $L^q$ norms and sign distribution, proven via interpolation and duality, with applications to Fourier, Hankel, and Dini series, and connections to sphere packing via linear programming.

ABSTRACT

We prove new sign uncertainty principles which vastly generalize the recent developments of Bourgain, Clozel & Kahane and Cohn & Gonçalves, and apply our results to a variety of spaces and operators. In particular, we establish new sign uncertainty principles for Fourier and Dini series, the Hilbert transform, the discrete Fourier and Hankel transforms, spherical harmonics, and Jacobi polynomials, among others. We present numerical evidence highlighting the relationship between the discrete and continuous sign uncertainty principles for the Fourier and Hankel transforms, which in turn are connected with the sphere packing problem via linear programming. Finally, we explore some connections between the sign uncertainty principle on the sphere and spherical designs.

Motivation & Objective

• To generalize the recent sign uncertainty principles of Bourgain–Clozel–Kahane and Cohn–Gonçalves to a broad class of operators and function spaces.
• To establish a universal inequality that quantifies the trade-off between the size of the negative support of a function and its transform under general operators.
• To apply the new principle to diverse settings including Fourier and Dini series, Hilbert and Hankel transforms, spherical harmonics, and Jacobi polynomials.
• To explore numerical connections between discrete and continuous sign uncertainty principles and their relevance to the sphere packing problem via linear programming.
• To investigate links between sign uncertainty on the sphere and spherical designs, particularly in the context of extremal functions.

Proposed method

• Derive a general operator sign uncertainty principle using interpolation and duality in $L^p$ spaces, with constraints on $L^1$, $L^p$, and $L^q$ norms.
• Define the family $\mathcal{F}$ of function pairs $(f,g)$ linked by an invertible operator $T$, with sign and integral constraints: $\int f \leq 0$, $\int g \leq 0$.
• Apply Hölder's inequality and conjugate exponent relations ($p'$, $q'$) to derive the key lower bound on the product of negative support measures.
• Use the Gurobi solver with PARI/GP interface to compute numerical feasibility for discrete Hankel and Fourier transforms.
• Establish connections between sign uncertainty and linear programming bounds by analyzing feasible parameter pairs $(k, q_\pm)$ in tables.
• Leverage symmetry and duality to extend results to spherical harmonics and Jacobi polynomials via known orthogonal function theory.

Experimental results

Research questions

• RQ1 Can the sign uncertainty principle be generalized beyond the Fourier transform to arbitrary operators and function spaces?
• RQ2 What is the minimal measure of the negative support of a function and its transform under general $L^p$-bounded operators?
• RQ3 How do discrete and continuous sign uncertainty principles relate, particularly in the context of sphere packing via linear programming?
• RQ4 What is the role of spherical designs in the sign uncertainty principle on the sphere?
• RQ5 Can the new principle be used to derive new bounds for Dini series, Hankel transforms, and Gegenbauer polynomials?

Key findings

• The paper establishes a new universal sign uncertainty principle: for any nonzero $(f,g) \in \mathcal{F}$, the product of the $L^{p'}$-norm of the negative set of $f$ and the $L^q$-norm of the negative set of $g$ is bounded below by $a^{-1} b^{-q'/q} (2c)^{-q'}$.
• Numerical tables show that for the discrete Fourier transform, the $(-1, \frac{d}{2}-1)$-feasible pairs $(k, q_-)$ grow with dimension $d$, with $q_-$ values decreasing as $d$ increases, indicating tighter bounds in higher dimensions.
• For the discrete Hankel transform, the $(-1, \frac{d}{2}-1)$-feasible $q_-$ values decrease with increasing $d$, and the tables confirm feasibility for $k \leq 30$, $d \leq 30$, with $q_-$ dropping from 3 to 10 as $k$ increases.
• The $ (+1, \frac{d}{2}-1) $-feasible $q_+$ values in Table 3 show a similar trend: $q_+$ decreases with increasing $k$ and $d$, indicating that the uncertainty bound tightens for larger $k$ and $d$.
• The authors demonstrate that the continuous and discrete sign uncertainty principles are numerically connected, with the discrete case mirroring the continuous one, suggesting a deeper structural link.
• The results connect sign uncertainty to the sphere packing problem via linear programming, as the feasibility of $(-1, \frac{d}{2}-1)$-pairs corresponds to known bounds in the Cohn–Elkies linear programming framework.